Inverse Scattering for Singular Potentials in Two Dimensions

We consider the Schrödinger equation for a compactly supported potential having jump type singularities at a subdomain of ${\mathbb {R}^2}$. We prove that knowledge of the scattering amplitude at a fixed energy, determines the location of the singularity as well as the jump across the curve of discontinuity. This result follows from a similar result for the Dirichlet to Neumann map associated to the Schrödinger equation for a compactly supported potential with the same type of singularities.